Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 701section, the twistor–D operator is given byD ρ αf = X ρ R ′Y A α ∇ R′A f + wξ ρ αf, (4.179)where f is any weighted twistor-spinor object. Using this and the expressions(4.176), (4.177) for the twistor connection, the following identities areeasily established:DαX ρ β C = ′ Xρ C ′λβ K Y α K DαY ρ β C = −Yα C X ρ K ′ξK′ βD ρ αξ S′β= P ρS′αβD ρ αλ σ B = −P ρσαBX α B ′Dγ αf = wX γ B ′fY Bγ D γ αf = 0ξ B′γ D γ αf = D B′α f λ α BD B′α f = ∇ B′B f,where, again, f is any weighted twistor-spinor and we writeP ρσαβ:= P R′ S ′AB Xρ R ′Xσ S ′Y α A Yβ B, P ρS′αβ:= P R′ S ′AB Xρ R ′Y α A Yβ B, P ρσP R′ S ′ABXρ R ′Xσ S ′Y A α , etc.(4.180)αB :=Notice also that the objects ξ B′α and λ β Adescribing the splitting of thetwistors can be viewed as the projecting parts of ξ β α := ξ −1 Dαξ β andδ β α − ξ β α, respectively.D−CurvatureFor f ∈ E[w] the projecting part of Dαf ρ is 1 p Xα ′P ′DP α f = wf. Althoughthis is 0th order in f, this part of Dαf ρ behaves like a first order operatorbecause of the weight factor, w. In particular 1 p Xα ′P ′DP α satisfies a Leibnizrule and so therefore so does Dα. ρ It follows immediately that, acting onE µ [w], [Dα, ρ Dβ σ ] decomposes into a 0th order curvature part and a 1st ordertorsion part. In fact it is easy using the identities (4.171) and (4.180) toverify that[D ρ α, D σ β]v µ = W ρσµαβγ vγ − W ρσναβγ Dγ ν v µ + δ σ αD ρ β vµ − δ ρ β Dσ αv µ ,whereW ρσµαβγ = Xρ A ′Xσ B ′Y A α Y B β W A′ B ′ µABγ .4.13.4 Application: Rovelli’s Loop Quantum Gravity4.13.4.1 Introduction to Loop Quantum GravityRecall (from subsection 3.10.4 above) that Carlo Rovelli developed (in thelast decade of the 20th Century) the so–called loop approach to quantum